Abstract
We study the supersymmetric Casimir energy E susy of \( \mathcal{N}=1 \) field theories with an R-symmetry, defined on rigid supersymmetric backgrounds S 1 ×M 3, using a Hamiltonian formalism. These backgrounds admit an ambi-Hermitian geometry, and we show that the net contributions to E susy arise from certain twisted holomorphic modes on ℝ × M 3, with respect to both complex structures. The supersymmetric Casimir energy may then be identified as a limit of an index-character that counts these modes. In particular this explains a recent observation relating E susy on S 1 × S 3 to the anomaly polynomial. As further applications we compute E susy for certain secondary Hopf surfaces, and discuss how the index-character may also be used to compute generalized supersymmetric indices.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
B. Assel, D. Cassani and D. Martelli, Localization on Hopf surfaces, JHEP 08 (2014) 123 [arXiv:1405.5144] [INSPIRE].
J. Lorenzen and D. Martelli, Comments on the Casimir energy in supersymmetric field theories, JHEP 07 (2015) 001 [arXiv:1412.7463] [INSPIRE].
B. Assel et al., The Casimir energy in curved space and its supersymmetric counterpart, JHEP 07 (2015) 043 [arXiv:1503.05537] [INSPIRE].
N. Bobev, M. Bullimore and H.-C. Kim, Supersymmetric Casimir energy and the anomaly polynomial, JHEP 09 (2015) 142 [arXiv:1507.08553] [INSPIRE].
C. Klare, A. Tomasiello and A. Zaffaroni, Supersymmetry on curved spaces and holography, JHEP 08 (2012) 061 [arXiv:1205.1062] [INSPIRE].
T.T. Dumitrescu, G. Festuccia and N. Seiberg, Exploring curved superspace, JHEP 08 (2012) 141 [arXiv:1205.1115] [INSPIRE].
D. Martelli, J. Sparks and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation, Commun. Math. Phys. 280 (2008) 611 [hep-th/0603021] [INSPIRE].
C. Romelsberger, Counting chiral primaries in N = 1, D = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
C. Closset and I. Shamir, The \( \mathcal{N}=1 \) chiral multiplet on T 2 × S 2 and supersymmetric localization, JHEP 03 (2014) 040 [arXiv:1311.2430] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, The geometry of supersymmetric partition functions, JHEP 01 (2014) 124 [arXiv:1309.5876] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, Supersymmetric field theories on three-manifolds, JHEP 05 (2013) 017 [arXiv:1212.3388] [INSPIRE].
L.F. Alday, D. Martelli, P. Richmond and J. Sparks, Localization on three-manifolds, JHEP 10 (2013) 095 [arXiv:1307.6848] [INSPIRE].
E. Lerman and S. Tolman, Symplectic toric orbifolds, Trans. Amer. Math. Soc. 349 (1997) 4201 [dg-ga/9412005].
T. Nishioka and I. Yaakov, Generalized indices for \( \mathcal{N}=1 \) theories in four-dimensions, JHEP 12 (2014) 150 [arXiv:1407.8520] [INSPIRE].
L. Di Pietro and Z. Komargodski, Cardy formulae for SUSY theories in d = 4 and d = 6, JHEP 12 (2014) 031 [arXiv:1407.6061] [INSPIRE].
E. Gerchkovitz, Constraints on the R-charges of free bound states from the Römelsberger index, JHEP 07 (2014) 071 [arXiv:1311.0487] [INSPIRE].
B. Assel, D. Cassani and D. Martelli, Supersymmetric counterterms from new minimal supergravity, JHEP 11 (2014) 135 [arXiv:1410.6487] [INSPIRE].
M. Spreafico, On the Barnes double zeta and Gamma functions, J. Numb. Theor. 129 (2009) 2035.
J.P. Gauntlett, D. Martelli, J. Sparks and S.-T. Yau, Obstructions to the existence of Sasaki-Einstein metrics, Commun. Math. Phys. 273 (2007) 803 [hep-th/0607080] [INSPIRE].
C. Romelsberger, Calculating the superconformal index and Seiberg duality, arXiv:0707.3702 [INSPIRE].
H.-C. Kim and S. Kim, M 5-branes from gauge theories on the 5-sphere, JHEP 05 (2013) 144 [arXiv:1206.6339] [INSPIRE].
F. Benini, T. Nishioka and M. Yamazaki, 4D index to 3D index and 2D TQFT, Phys. Rev. D 86 (2012) 065015 [arXiv:1109.0283] [INSPIRE].
S.S. Razamat and B. Willett, Global properties of supersymmetric theories and the Lens space, Commun. Math. Phys. 334 (2015) 661 [arXiv:1307.4381] [INSPIRE].
L.F. Alday, M. Fluder and J. Sparks, The large-N limit of M 2-branes on Lens spaces, JHEP 10 (2012) 057 [arXiv:1204.1280] [INSPIRE].
V.P. Spiridonov, Modified elliptic Gamma functions and 6d superconformal indices, Lett. Math. Phys. 104 (2014) 397 [arXiv:1211.2703] [INSPIRE].
M. Nishizawa, An elliptic analogue of the multiple Gamma function, J. Phys. A 34 (2001) 7411.
D. Cassani and D. Martelli, The gravity dual of supersymmetric gauge theories on a squashed S 1 × S 3, JHEP 08 (2014) 044 [arXiv:1402.2278] [INSPIRE].
D. Cassani, J. Lorenzen and D. Martelli, Comments on supersymmetric solutions of minimal gauged supergravity in five dimensions, Class. Quant. Grav. 33 (2016) 115013 [arXiv:1510.01380] [INSPIRE].
P.B. Genolini, D. Cassani, D. Martelli and J. Sparks, The holographic supersymmetric Casimir energy, arXiv:1606.02724 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1512.02521
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Martelli, D., Sparks, J. The character of the supersymmetric Casimir energy. J. High Energ. Phys. 2016, 117 (2016). https://doi.org/10.1007/JHEP08(2016)117
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2016)117